Non-conservative systems with symmetrizable stiffness matrices exhibiting limit cycles

Abstract

Non-conservative dissipative systems under partial follower loading, with stiffness matrices that are symmetrizable, are reexamined with the aid of non-linear analysis. In this work, the conditions under which the above autonomous systems may experience a limit cycle response, stable (periodic attractor) or unstable (dynamic instability via flutter) in a certain region of existence of adjacent equilibria (region of divergence instability) are properly established. This region where a limit cycle response may occur is defined by an interval of values of the non-conservativeness loading parameter η with lower bound η = η 0 (boundary between existence and non-existence of adjacent equilibria) and upper bound η = 0.50 (being invariant with respect to all other parameters). In this region although the set of buckling eigenvectors is complete (associated with distinct eigenvalues) there is one postbuckling equilibrium path passing through two consecutive branching points. Hence, only non-conservative systems with η > 0.50 (having all postbuckling equilibrium paths independent of each other) behave dynamically like symmetric (conservative) systems. These findings are verified with the aid of various numerical examples.